A016945 -id:A016945 - cp's OEIS frontend

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

A141419 Triangle read by rows: T(n, k) = A000217(n) - A000217(n - k) with 1 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 4, 7, 9, 10, 5, 9, 12, 14, 15, 6, 11, 15, 18, 20, 21, 7, 13, 18, 22, 25, 27, 28, 8, 15, 21, 26, 30, 33, 35, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55

Offset: 1

Roger L. Bagula, Aug 05 2008

As a rectangle, the accumulation array of A051340.
From Clark Kimberling, Feb 05 2011: (Start)
Here all the weights are divided by two where they aren't in Cahn.
As a rectangle, A141419 is in the accumulation chain
... < A051340 < A141419 < A185874 < A185875 < A185876 < ...
(See A144112 for the definition of accumulation array.)
row 1: A000027
col 1: A000217
diag (1,5,...): A000326 (pentagonal numbers)
diag (2,7,...): A005449 (second pentagonal numbers)
diag (3,9,...): A045943 (triangular matchstick numbers)
diag (4,11,...): A115067
diag (5,13,...): A140090
diag (6,15,...): A140091
diag (7,17,...): A059845
diag (8,19,...): A140672
(End)
Let N=2*n+1 and k=1,2,...,n. Let A_{N,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n](1,k), that is, the n-th row of the triangle is given by the first row of M_n. - _L. Edson Jeffery, Nov 20 2011
Conjecture. Let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x) = 1, U_1(x) = 2*x and U_r(x) = 2*x*U_(r-1)(x) - U_(r-2)(x)  (r>1). Define the column vectors V_(k-1) = (U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n-1)] be the n X n matrix formed by taking V_(k-1) as column k-1. Let X_N = [S_N]^T*S_N, and let [X_N](i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then t(n,k) = [X_N](k-1,k-1), and row n of the triangle is given by the main diagonal entries of X_N. Remarks: Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}] is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery, Jan 20 2012
Row sums: n*(n+1)*(2*n+1)/6. - L. Edson Jeffery, Jan 25 2013
n-th row = partial sums of n-th row of A004736. - Reinhard Zumkeller, Aug 04 2014
T(n,k) is the number of distinct sums made by at most k elements in {1, 2, ... n}, for 1 <= k <= n, e.g., T(6,2) = the number of distinct sums made by at most 2 elements in {1,2,3,4,5,6}. The sums range from 1, to 5+6=11. So there are 11 distinct sums. - Derek Orr, Nov 26 2014
A number n occurs in this sequence A001227(n) times, the number of odd divisors of n, see A209260. - Hartmut F. W. Hoft, Apr 14 2016
Conjecture: 2*n + 1 is composite if and only if gcd(t(n,m),m) != 1, for some m. - L. Edson Jeffery, Jan 30 2018
From Peter Munn, Aug 21 2019 in respect of the sequence read as a triangle: (Start)
A number m can be found in column k if and only if A286013(m, k) is nonzero, in which case m occurs in column k on row A286013(m, k).
The first occurrence of m is in row A212652(m) column A109814(m), which is the rightmost column in which m occurs. This occurrence determines where m appears in A209260. The last occurrence of m is in row m column 1.
Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A270877; this is the only one that contains its own row numbers only once.
(End)

			As a triangle:
   1,
   2,  3,
   3,  5,  6,
   4,  7,  9, 10,
   5,  9, 12, 14, 15,
   6, 11, 15, 18, 20, 21,
   7, 13, 18, 22, 25, 27, 28,
   8, 15, 21, 26, 30, 33, 35, 36,
   9, 17, 24, 30, 35, 39, 42, 44, 45,
  10, 19, 27, 34, 40, 45, 49, 52, 54, 55;
As a rectangle:
   1   2   3   4   5   6   7   8   9  10
   3   5   7   9  11  13  15  17  19  21
   6   9  12  15  18  21  24  27  30  33
  10  14  18  22  26  30  34  38  42  46
  15  20  25  30  35  40  45  50  55  60
  21  27  33  39  45  51  57  63  69  75
  28  35  42  49  56  63  70  77  84  91
  36  44  52  60  68  76  84  92 100 108
  45  54  63  72  81  90  99 108 117 126
  55  65  75  85  95 105 115 125 135 145
Since the odd divisors of 15 are 1, 3, 5 and 15, number 15 appears four times in the triangle at t(3+(5-1)/2, 5) in column 5 since 5+1 <= 2*3, t(5+(3-1)/2, 3), t(1+(15-1)/2, 2*1) in column 2 since 15+1 > 2*1, and t(15+(1-1)/2, 1). - _Hartmut F. W. Hoft_, Apr 14 2016

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.
Carlton Gamer, David W. Roeder, and John J. Watkins, Trapezoidal Numbers, Mathematics Magazine 58:2 (1985), pp. 108-110.
L. E. Jeffery, Unit-primitive matrices
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000330 (row sums), A004736, A057059, A070543.
A144112, A051340, A141419, A185874, A185875, A185876 are accumulation chain related.
A141418 is a variant.
Cf. A001227, A209260. - Hartmut F. W. Hoft, Apr 14 2016
A109814, A212652, A270877, A286013 relate to where each natural number appears in this sequence.
A000027, A000217, A000326, A005449, A045943, A059845, A115067, A140090, A140091, A140672 are rows, columns or diagonals - refer to comments.

a141419 n k =  k * (2 * n - k + 1) `div` 2
a141419_row n = a141419_tabl !! (n-1)
a141419_tabl = map (scanl1 (+)) a004736_tabl
-- Reinhard Zumkeller, Aug 04 2014

a:=(n,k)->k*n-binomial(k,2): seq(seq(a(n,k),k=1..n),n=1..12); # Muniru A Asiru, Oct 14 2018

T[n_, m_] = m*(2*n - m + 1)/2; a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a]

t(n,m) = m*(2*n - m + 1)/2.
t(n,m) = A000217(n) - A000217(n-m). - L. Edson Jeffery, Jan 16 2013
Let v = d*h with h odd be an integer factorization, then v = t(d+(h-1)/2, h) if h+1 <= 2*d, and v = t(d+(h-1)/2, 2*d) if h+1 > 2*d; see A209260. - Hartmut F. W. Hoft, Apr 14 2016
G.f.: y*(-x + y)/((-1 + x)^2*(-1 + y)^3). - Stefano Spezia, Oct 14 2018
T(n, 2) = A060747(n) for n > 1. T(n, 3) = A008585(n - 1) for n > 2. T(n, 4) = A016825(n - 2) for n > 3. T(n, 5) = A008587(n - 2) for n > 4. T(n, 6) = A016945(n - 3) for n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7.r n > 5. T(n, 7) = A008589(n - 3) for n > 6. T(n, 8) = A017113(n - 4) for n > 7. T(n, 9) = A008591(n - 4) for n > 8. T(n, 10) = A017329(n - 5) for n > 9. T(n, 11) = A008593(n - 5) for n > 10.  T(n, 12) = A017593(n - 6) for n > 11. T(n, 13) = A008595(n - 6) for n > 12. T(n, 14) = A147587(n - 7) for n > 13. T(n, 15) = A008597(n - 7) for n > 14. T(n, 16) = A051062(n - 8) for n > 15. T(n, 17) = A008599(n - 8) for n > 16. - Stefano Spezia, Oct 14 2018
T(2*n-k, k) = A070543(n, k). - Peter Munn, Aug 21 2019

Simpler name by Stefano Spezia, Oct 14 2018

A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 20, 21, 22, 25, 24, 19, 26, 27, 28, 29, 30, 37, 32, 33, 34, 35, 36, 41, 38, 39, 40, 43, 42, 47, 44, 45, 46, 53, 48, 31, 50, 51, 52, 61, 54, 49, 56, 57, 58, 67, 60, 71, 62, 63, 64, 65, 66, 77, 68, 69, 70, 83, 72, 89, 74, 75, 76, 59, 78, 91, 80, 81

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

a(n) tells which number in Ludic array A255127 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A255129 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

			A083221(8,1) = 19 and A255127(8,1) = 23, thus a(19) = 23.
A083221(9,1) = 23 and A255127(9,1) = 25, thus a(23) = 25.
A083221(3,2) = 25 and A255127(3,2) = 19, thus a(25) = 19.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Cf. A001248, A003309, A007310, A008578, A083221, A084967, A252460, A254100, A255413, A255127.
Inverse: A255408.
Similar permutations: A249818.

a(n) = A255127(A252460(n)).
Other identities. For all n >= 1:
a(2n) = 2n. [Fixes even numbers.]
a(3n) = 3n. [Fixes multiples of three.]
a(A008578(n)) = A003309(n). [Maps noncomposites to Ludic numbers.]
a(A001248(n)) = A254100(n). [Maps squares of primes to "postludic numbers".]
a(A084967(n)) = a(5*A007310(n)) = A007310((5*n)-3) = A255413(n). [Maps A084967 to A255413.]
(And similarly between other columns and rows of A083221 and A255127.)

A016946 a(n) = (6*n+3)^2.

Original entry on oeis.org

9, 81, 225, 441, 729, 1089, 1521, 2025, 2601, 3249, 3969, 4761, 5625, 6561, 7569, 8649, 9801, 11025, 12321, 13689, 15129, 16641, 18225, 19881, 21609, 23409, 25281, 27225, 29241, 31329, 33489, 35721, 38025, 40401, 42849, 45369, 47961, 50625, 53361, 56169

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A002378, A006752, A016754, A016945, A086729.

[(6*n+3)^2: n in [0..60]]; // Vincenzo Librandi, May 05 2011

A016946:=n->(6*n+3)^2: seq(A016946(n), n=0..50); # Wesley Ivan Hurt, Oct 13 2014

(6 Range[0, 50] + 3)^2 (* or *)
CoefficientList[Series[9 (1 + 6 x + x^2)/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Oct 13 2014 *)
LinearRecurrence[{3,-3,1},{9,81,225},40] (* Harvey P. Dale, Jul 13 2015 *)

a(n)=(6*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 36*A002378(n)+9. - Jean-Bernard François, Oct 12 2014
From Wesley Ivan Hurt, Oct 13 2014: (Start)
G.f.: 9*(1+6*x+x^2)/(1-x)^3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).
a(n) = A016945(n)^2 = A000290(A016945(n)). (End)
Sum_{n>=0} 1/a(n) = A086729. - Amiram Eldar, Nov 16 2020
a(n) = 9*A016754(n). - R. J. Mathar, Dec 11 2020
Sum_{n>=0} (-1)^n/a(n) = G/9, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 30 2022
E.g.f.: 9*exp(x)*(1 + 8*x + 4*x^2). - Stefano Spezia, Aug 19 2022

A047270 Numbers that are congruent to {3, 5} mod 6.

Original entry on oeis.org

3, 5, 9, 11, 15, 17, 21, 23, 27, 29, 33, 35, 39, 41, 45, 47, 51, 53, 57, 59, 63, 65, 69, 71, 75, 77, 81, 83, 87, 89, 93, 95, 99, 101, 105, 107, 111, 113, 117, 119, 123, 125, 129, 131, 135, 137, 141, 143, 147, 149

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 10 ).

This sequence is an interleaving of A016945 with A016969. - Guenther Schrack, Nov 16 2018

Bruno Berselli, Table of n, a(n) for n = 1..10000 (From _Bruno Berselli_, Jun 24 2010)
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047235 [(6*n-(-1)^n-3)/2], A047241 [(6*n-(-1)^n-5)/2], A047238 [(6*n-(-1)^n-7)/2]. [Bruno Berselli, Jun 24 2010]
Subsequence of A186422.
From Guenther Schrack, Nov 18 2018: (Start)
Complement: A047237.
First differences: A105397(n) for n > 0.
Partial sums: A227017(n+1) for n > 0.
Elements of odd index: A016945.
Elements of even index: A016969(n-1) for n > 0. (End)

Select[Range@ 149, MemberQ[{3, 5}, Mod[#, 6]] &] (* or *)
Array[(6 # - (-1)^# - 1)/2 &, 50] (* or *)
Fold[Append[#1, 6 #2 - Last@ #1 - 4] &, {3}, Range[2, 50]] (* or *)
CoefficientList[Series[(3 + 2 x + x^2)/((1 + x) (1 - x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jan 12 2018 *)

a(n) = (6*n - 1 - (-1)^n)/2 \\ David Lovler, Aug 25 2022

a(n) = sqrt(2)*sqrt((1-6*n)*(-1)^n + 18*n^2 - 6*n + 1)/2. - Paul Barry, May 11 2003
From Bruno Berselli, Jun 24 2010: (Start)
G.f.: (3+2*x+x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0, with n > 3.
a(n) = (6*n - (-1)^n - 1)/2. (End)
a(n) = 6*n - a(n-1) - 4 with n > 1, a(1)=3. - Vincenzo Librandi, Aug 05 2010
From Guenther Schrack, Nov 17 2018: (Start)
a(n) = a(n-2) + 6 for n > 2.
a(-n) = -A047241(n+1) for n > 0.
a(n) = A109613(n-1) + 2*n for n > 0.
a(n) = 2*A001651(n) + 1.
m-element moving averages: Sum_{k=1..m} a(n-m+k)/m = A016777(n-m/2) for m = 2, 4, 6, ... and n >= m. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) - log(3)/4. - Amiram Eldar, Dec 13 2021
E.g.f.: 1 + 3*x*exp(x) - cosh(x). - David Lovler, Aug 25 2022

A108701 Values of n such that n^2-2 and n^2+2 are both prime.

Original entry on oeis.org

3, 9, 15, 21, 33, 117, 237, 273, 303, 309, 387, 429, 441, 447, 513, 561, 573, 609, 807, 897, 1035, 1071, 1113, 1143, 1233, 1239, 1311, 1563, 1611, 1617, 1737, 1749, 1827, 1839, 1953, 2133, 2211, 2283, 2589, 2715, 2721, 2955, 3081, 3093, 3453, 3549, 3555, 3621, 3723, 3807

Offset: 1

John L. Drost, Jun 19 2005

Since x^2 + 2 is divisible by 3 unless x is divisible by 3, all elements are 3 mod 6.

Intersection of A067201 and A028870. - Robert Israel, Sep 11 2014

			21 is on the list since 21^2 - 2 = 439 and 21^2 + 2 = 443 are primes.

David Wells, Prime Numbers, John Wiley and Sons, 2005, p. 219 (article:'Siamese primes')

Nathaniel Johnston, Table of n, a(n) for n = 1..8750
Raymond A. Beauregard and E. R. Suryanarayan, Square-plus-two primes, Mathematical Gazette 85(502) 90-1.

Subsequence of A016945.

Cf. A016945, A028870, A028873, A038599, A067201, A153974.

[n: n in [3..3600 by 6] | IsPrime(n^2-2) and IsPrime(n^2+2)];  // Bruno Berselli, Apr 15 2011

select(n -> isprime(n^2-2) and isprime(n^2+2), [seq(6*i+3,i=0..1000)]); # Robert Israel, Sep 11 2014

Select[Range[5000], PrimeQ[#^2 - 2] && PrimeQ[#^2 + 2] &] (* Alonso del Arte, Sep 11 2014 *)

is(n)=isprime(n^2-2)&&isprime(n^2+2) \\ Charles R Greathouse IV, Jul 02 2013

Terms corrected by Charles R Greathouse IV, Sep 11 2014

A048701 List of binary palindromes of even length (written in base 10).

Original entry on oeis.org

0, 3, 9, 15, 33, 45, 51, 63, 129, 153, 165, 189, 195, 219, 231, 255, 513, 561, 585, 633, 645, 693, 717, 765, 771, 819, 843, 891, 903, 951, 975, 1023, 2049, 2145, 2193, 2289, 2313, 2409, 2457, 2553, 2565, 2661, 2709, 2805, 2829, 2925, 2973, 3069, 3075, 3171, 3219, 3315

Offset: 0

Antti Karttunen, Mar 07 1999

A178225(a(n)) = 1. - Reinhard Zumkeller, Oct 21 2011

a(n) is divisible by 3 and it is always an odd number for n > 0. Therefore a(n) is in A016945 for n > 0. - Altug Alkan, Dec 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

See also A048702 = this sequence divided by 3, A048700 = binary palindromes of odd length, A006995 = all binary palindromes, A048703 = quaternary (base 4) palindromes of even length.
For first differences see A265026, A265027.
Cf. A030308, A007088, A178225.

a048701 n = foldr (\d v -> 2 * v + d) 0 (reverse bs ++ bs) where
   bs = a030308_row (n)
-- Reinhard Zumkeller, Feb 19 2003, Oct 21 2011

Prepend[Select[Range@ 3315, Reverse@ # == # && EvenQ@ Length@ # &@ IntegerDigits[#, 2] &], 0] (* Michael De Vlieger, Dec 04 2015 *)

a048701(n) = my(f); f = length(binary(n)) - 1; 2^(f+1)*n + sum(i=0, f, bittest(n, i) * 2^(f-i)); \\ Altug Alkan, Dec 03 2015

def A048701(n):
    s = bin(n)[2:]
    return int(s+s[::-1],2) # Chai Wah Wu, Feb 26 2021

a(n) = (2^(floor_log_2(n)+1))*n + Sum_{i=0..floor_log_2(n)} '(bit_i(n, i)*(2^(floor_log_2(n)-i)))'.

Offset corrected by Reinhard Zumkeller, Oct 21 2011

Offset changed back to 0 by Andrey Zabolotskiy, Dec 26 2022

A260738 Row index to A255127: a(1) = 0; for n > 1, a(n) = number of the stage where n is removed in the sieve which produces Ludic numbers.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 1, 3, 1, 2, 1, 8, 1, 9, 1, 2, 1, 10, 1, 4, 1, 2, 1, 3, 1, 11, 1, 2, 1, 12, 1, 13, 1, 2, 1, 14, 1, 3, 1, 2, 1, 15, 1, 5, 1, 2, 1, 4, 1, 16, 1, 2, 1, 3, 1, 17, 1, 2, 1, 18, 1, 6, 1, 2, 1, 19, 1, 3, 1, 2, 1, 20, 1, 4, 1, 2, 1, 21, 1, 22, 1, 2, 1, 3, 1, 23, 1, 2, 1, 7, 1, 5, 1, 2, 1, 24, 1, 3, 1, 2, 1, 4, 1, 25, 1, 2, 1, 26, 1

Offset: 1

Antti Karttunen, Jul 30 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10105

Row index to array A255127.
Cf. A003309, A254100.
Cf. A260739 (corresponding column index).
Cf. A055396, A260438 for row indices to other arrays similar to A255127.
Differs from A055396 for the first time at n=19.

(define (A260738 n) (cond ((= 1 n) 0) ((even? n) 1) (else (let searchrow ((row 2)) (let searchcol ((col 1)) (cond ((>= (A255127bi row col) n) (if (= (A255127bi row col) n) row (searchrow (+ 1 row)))) (else (searchcol (+ 1 col))))))))) ;; Code for A255127bi given in A255127.

Other identities. For all n >= 1:
a(A003309(n)) = n-1. [In Ludic sieve A003309(k+1) (i.e., the k-th Ludic number after 1) is the first among the numbers removed at stage k.]
a(2n) = 1. [All even numbers are removed at the stage one of the sieve.]
a(A016945(n)) = 2, a(A255413(n)) = 3, a(A255414(n)) = 4, ..., a(A255419(n)) = 9.
a(A254100(n)) = n.
For all n >= 2:
A255127(a(n), A260739(n)) = n.

A016947 a(n) = (6*n + 3)^3.

Original entry on oeis.org

27, 729, 3375, 9261, 19683, 35937, 59319, 91125, 132651, 185193, 250047, 328509, 421875, 531441, 658503, 804357, 970299, 1157625, 1367631, 1601613, 1860867, 2146689, 2460375, 2803221, 3176523, 3581577, 4019679, 4492125, 5000211, 5545233, 6128487, 6751269

Offset: 0

N. J. A. Sloane

			a(0) = (6*0 + 3)^3 = 3^3 = 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A016755, A016911, A016923, A016935, A016945, A016946, A016959, A016971.

[(6*n+3)^3: n in [0..50]]; // Vincenzo Librandi, May 05 2011

Table[(6*n + 3)^3, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020 *)
LinearRecurrence[{4,-6,4,-1},{27,729,3375,9261},40] (* Harvey P. Dale, Jul 02 2025 *)

Sum_{n>=0} 1/a(n) = 7*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
G.f.: 27*(1+x)*(1+22*x+x^2)/(-1+x)^4. - Wesley Ivan Hurt, Oct 02 2020
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^3.
a(n) = 3^3*A016755(n).
Sum_{n>=0} (-1)^n/a(n) = Pi^3/864. (End)

cp's OEIS Frontend

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

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A141419 Triangle read by rows: T(n, k) = A000217(n) - A000217(n - k) with 1 <= k <= n.

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A255407 Permutation of natural numbers: a(n) = A255127(A252460(n)).

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A016946 a(n) = (6*n+3)^2.

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A047270 Numbers that are congruent to {3, 5} mod 6.

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A108701 Values of n such that n^2-2 and n^2+2 are both prime.

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A048701 List of binary palindromes of even length (written in base 10).

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