A108195 -id:A108195 - cp's OEIS frontend

A176271 The odd numbers as a triangle read by rows.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Reinhard Zumkeller, Apr 13 2010

A108309(n) = number of primes in n-th row.

			From _Philippe Deléham_, Oct 03 2011: (Start)
Triangle begins:
   1;
   3,  5;
   7,  9, 11;
  13, 15, 17, 19;
  21, 23, 25, 27, 29;
  31, 33, 35, 37, 39, 41;
  43, 45, 47, 49, 51, 53, 55;
  57, 59, 61, 63, 65, 67, 69, 71;
  73, 75, 77, 79, 81, 83, 85, 87, 89; (End)

G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Nicomachus's Theorem
Wikipedia, Nikomachos von Gerasa

Cf. A000466, A000537, A000578, A001105, A002061, A005408, A014209.
Cf. A016754, A027688, A027690, A027692, A027694, A028387, A048058.
Cf. A053755, A065599, A082111, A108195, A108309, A214604, A214661.

a176271 n k = a176271_tabl !! (n-1) !! (k-1)
a176271_row n = a176271_tabl !! (n-1)
a176271_tabl = f 1 a005408_list where
   f x ws = us : f (x + 1) vs where (us, vs) = splitAt x ws
-- Reinhard Zumkeller, May 24 2012

[n^2-n+2*k-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

A176271 := proc(n,k)
    n^2-n+2*k-1 ;
end proc: # R. J. Mathar, Jun 28 2013

Table[n^2-n+2*k-1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[n^2-n+2*k-1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = n^2 - n + 2*k - 1 for 1 <= k <= n.
T(n, k) = A005408(n*(n-1)/2 + k - 1).
T(2*n-1, n) = A016754(n-1) (main diagonal).
T(2*n, n) = A000466(n).
T(2*n, n+1) = A053755(n).
T(n, k) + T(n, n-k+1) = A001105(n), 1 <= k <= n.
T(n, 1)   = A002061(n), central polygonal numbers.
T(n, 2)   = A027688(n-1) for n > 1.
T(n, 3)   = A027690(n-1) for n > 2.
T(n, 4)   = A027692(n-1) for n > 3.
T(n, 5)   = A027694(n-1) for n > 4.
T(n, 6)   = A048058(n-1) for n > 5.
T(n, n-3) = A108195(n-2) for n > 3.
T(n, n-2) = A082111(n-2) for n > 2.
T(n, n-1) = A014209(n-1) for n > 1.
T(n, n) = A028387(n-1).
Sum_{k=1..n} T(n, k) = A000578(n) (Nicomachus's theorem).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A065599(n) (alternating sign row sums).
Sum_{j=1..n} (Sum_{k=1..n} T(j, k)) = A000537(n) (sum of first n rows).

A156140 Accumulation of Stern's diatomic series: a(0)=-1, a(1)=0, and a(n+1) = (2e(n)+1)*a(n) - a(n-1) for n > 1, where e(n) is the highest power of 2 dividing n.

Original entry on oeis.org

-1, 0, 1, 3, 2, 7, 5, 8, 3, 13, 10, 17, 7, 18, 11, 15, 4, 21, 17, 30, 13, 35, 22, 31, 9, 32, 23, 37, 14, 33, 19, 24, 5, 31, 26, 47, 21, 58, 37, 53, 16, 59, 43, 70, 27, 65, 38, 49, 11, 50, 39, 67, 28, 73, 45, 62, 17, 57, 40, 63, 23, 52, 29, 35, 6, 43, 37, 68, 31, 87, 56, 81, 25, 94, 69

Offset: 0

Arie Werksma (Werksma(AT)Tiscali.nl), Feb 04 2009

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A002487, A007814.
From Yosu Yurramendi, Mar 09 2018: (Start)
a(2^m +  0) = A000027(m),   m >= 0.
a(2^m +  1) = A002061(m+2), m >= 1.
a(2^m +  2) = A002522(m),   m >= 2.
a(2^m +  3) = A033816(m-1), m >= 2.
a(2^m +  4) = A002061(m),   m >= 2.
a(2^m +  5) = A141631(m),   m >= 3.
a(2^m +  6) = A084849(m-1), m >= 3.
a(2^m +  7) = A056108(m-1), m >= 3.
a(2^m +  8) = A000290(m-1), m >= 3.
a(2^m +  9) = A185950(m-1), m >= 4.
a(2^m + 10) = A144390(m-1), m >= 4.
a(2^m + 12) = A014106(m-2), m >= 4.
a(2^m + 16) = A028387(m-3), m >= 4.
a(2^m + 18) = A250657(m-4), m >= 5.
a(2^m + 20) = A140677(m-3), m >= 5.
a(2^m + 32) = A028872(m-2), m >= 5.
a(2^m -  1) = A005563(m-1), m >= 0.
a(2^m -  2) = A028387(m-2), m >= 2.
a(2^m -  3) = A033537(m-2), m >= 2.
a(2^m -  4) = A008865(m-1), m >= 3.
a(2^m -  7) = A140678(m-3), m >= 3.
a(2^m -  8) = A014209(m-3), m >= 4.
a(2^m - 16) = A028875(m-2), m >= 5.
a(2^m - 32) = A108195(m-5), m >= 6.
(End)

A156140 := proc(n)
    option remember ;
    if n <= 1 then
        n-1 ;
    else
        (2*A007814(n-1)+1)*procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A156140(n),n=0..80) ; # R. J. Mathar, Mar 14 2009

Fold[Append[#1, (2 IntegerExponent[#2, 2] + 1) #1[[-1]] - #1[[-2]] ] &, {-1, 0}, Range[73]] (* Michael De Vlieger, Mar 09 2018 *)

first(n)=my(v=vector(n+1)); v[1]=-1; v[2]=0; for(k=1,n-1,v[k+2]=(2*valuation(k,2)+1)*v[k+1] - v[k]); v \\ Charles R Greathouse IV, Apr 05 2016

fusc(n)=my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b
a(n)=my(m=1,s,t); if(n==0, return(-1)); while(n%2==0, s+=fusc(n>>=1)); while(n>1, t=logint(n,2); n-=2^t; s+=m*fusc(n)*(t^2+t+1); m*=-t); m*(n-1) + s \\ Charles R Greathouse IV, Dec 13 2016

a <- c(0,1)
maxlevel <- 6 # by choice
for(m in 1:maxlevel) {
  a[2^(m+1)] <- m + 1
  for(k in 1:(2^m-1)) {
    r <- m - floor(log2(k)) - 1
    a[2^r*(2*k+1)] <- a[2^r*(2*k)] + a[2^r*(2*k+2)]
}}
a
# Yosu Yurramendi, May 08 2018

Let b(n) = A002487(n), Stern's diatomic series.
a(n+1)*b(n) - a(n)*b(n+1) = 1 for n >= 0.
a(2n+1) = a(n) + a(n+1) + b(n) + b(n+1) for n >= 0.
a(2n) = a(n) + b(n) for n >= 0.
a(2^n + k) = -n*a(k) + (n^2 + n + 1)*b(k) for 0 <= k <= 2^n.
b(2^n + k) = -a(k) + (n + 1)*b(k) for 0 <= k <= 2^n.
a(2^m + k) = b(2^m+k)*m + b(k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 09 2018
a(2^(m+1)+2^m+1) = 2*m+1, m >= 0. - Yosu Yurramendi, Mar 09 2018
From Yosu Yurramendi, May 08 2018: (Start)
a(2^m) = m, m >= 0.
a(2^r*(2*k+1)) = a(2^r*(2*k)) + a(2^r*(2*k+2)), r = m - floor(log_2(k)) - 1, m > 0, 1 <= k < 2^m.
(End)

A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

Original entry on oeis.org

-1, -1, 0, -1, 1, 3, -1, 2, 5, 8, -1, 3, 7, 11, 15, -1, 4, 9, 14, 19, 24, -1, 5, 11, 17, 23, 29, 35, -1, 6, 13, 20, 27, 34, 41, 48, -1, 7, 15, 23, 31, 39, 47, 55, 63, -1, 8, 17, 26, 35, 44, 53, 62, 71, 80, -1, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, -1, 10, 21, 32, 43, 54, 65, 76, 87

Offset: 1

Jonathan Vos Post, Sep 13 2008

Arises in complete intersection threefolds,
Also can be produced as a triangle read by rows: a(n, k) = nk - (n + k). - Alonso del Arte, Jul 09 2009
Kosta: Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively. with n=>k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)*(n-1) - 1 singular points, then it is factorial.

			From _R. J. Mathar_, Jul 10 2009: (Start)
The rows A(n,1), A(n,2), A(n,3), etc., are :
.-1...0...3...8..15..24..35..48..63..80..99.120.143.168 A067998
.-1...1...5..11..19..29..41..55..71..89.109.131.155.181 A028387
.-1...2...7..14..23..34..47..62..79..98.119.142.167.194 A008865
.-1...3...9..17..27..39..53..69..87.107.129.153.179.207 A014209
.-1...4..11..20..31..44..59..76..95.116.139.164.191.220 A028875
.-1...5..13..23..35..49..65..83.103.125.149.175.203.233 A108195
.-1...6..15..26..39..54..71..90.111.134.159.186.215.246
.-1...7..17..29..43..59..77..97.119.143.169.197.227.259
.-1...8..19..32..47..64..83.104.127.152.179.208.239.272
.-1...9..21..35..51..69..89.111.135.161.189.219.251.285
.-1..10..23..38..55..74..95.118.143.170.199.230.263.298
.-1..11..25..41..59..79.101.125.151.179.209.241.275.311
.-1..12..27..44..63..84.107.132.159.188.219.252.287.324
.-1..13..29..47..67..89.113.139.167.197.229.263.299.337 Cf. A126719.
(End)
As a triangle:
. 0
. 1, 3
. 2, 5, 8
. 3, 7, 11, 15
. 4, 9, 14, 19, 24
. 5, 11, 17, 23, 29, 35
. 6, 13, 20, 27, 34, 41, 48
. 7, 15, 23, 31, 39, 47, 55, 63
. 8, 17, 26, 35, 44, 53, 62, 71, 80

Dimitra Kosta, Factoriality of complete intersection threefolds arXiv:0808.4071 [math.AG]

Row 1 = A067998(n) for n>0. Row 2 = A028387(n) for n>0.Column 1 = -A000012(n). Column 2 = A001477. Column 3 = A005408(k). Column 4 = A016789(k+1). Column 5 = A004767(k+2). Column 6 = A016897(k+3). Column 7 = A016969(k+4). Column 8 = A017053(k+5). Column 9 = A004771(k+6). Column 10 = A017257(k+7).

Cf. A000012, A001477, A004767, A004771, A005408, A016789, A016897, A016969, A017053, A028387, A067998, A126719.

A := proc(k,n) (n+k-2)*(n-1)-1 ; end: for d from 1 to 13 do for n from 1 to d do printf("%d,",A(d-n+1,n)) ; od: od: # R. J. Mathar, Jul 10 2009

a[n_, k_] := a[n, k] = n*k - (n + k); ColumnForm[Table[a[n, k], {n, 10}, {k, n}], Center] (* Alonso del Arte, Jul 09 2009 *)

A[k,n] = (n+k-2)*(n-1) - 1.

Antidiagonal sum: Sum_{n=1..d} A(d-n+1,n) = d*(d^2-2d-1)/2 = -A110427(d). - R. J. Mathar, Jul 10 2009

Duplicate of 6th antidiagonal removed by R. J. Mathar, Jul 10 2009
Keyword:tabl added by R. J. Mathar, Jul 23 2009
Edited by N. J. A. Sloane, Sep 14 2009. There was a comment that the defining formula was wrong, but it is perfectly correct.

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   9  16  25  36 ...
   7   8   6  11  18  27 ...
  13  14  12  15  24  35 ...
  21  22  20  23  19  28 ...
  31  32  30  33  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  9,  8, 13;
  17, 16,  6, 14, 21;
  26, 25, 11, 12, 22, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A108240 Numbers k such that (10^k)^2 + 5*(10^k) - 1 is prime.

Original entry on oeis.org

0, 1, 2, 7, 45, 54, 60, 99, 195, 271, 476, 1491, 2713, 8155

Offset: 1

Jason Earls, Jun 17 2005

All of the primes for the terms <= 2713 have been certified. Primality proof for the largest: PFGW Version 20041001.Win_Stable (v1.2 RC1b) [FFT v23.8] Primality testing (10^2713)^2+5*(10^2713)-1 [N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 7, base 1+sqrt(7) Calling Brillhart-Lehmer-Selfridge with factored part 34.96% (10^2713)^2+5*(10^2713)-1 is prime! (216.4510s+10.6592s)

Cf. A108195.

is(n)=ispseudoprime((10^n)^2+5*(10^n)-1) \\ Charles R Greathouse IV, Jun 13 2017

a(14) from Michael S. Branicky, Mar 27 2023

cp's OEIS Frontend

A176271 The odd numbers as a triangle read by rows.

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A156140 Accumulation of Stern's diatomic series: a(0)=-1, a(1)=0, and a(n+1) = (2e(n)+1)*a(n) - a(n-1) for n > 1, where e(n) is the highest power of 2 dividing n.

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A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

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A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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A108240 Numbers k such that (10^k)^2 + 5*(10^k) - 1 is prime.

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