A255437 -id:A255437 - cp's OEIS frontend

A255527 Where records occur in A255437.

1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Reinhard Zumkeller, Mar 23 2015

nonn

A255437(a(n)) = A164514(n) and A255437(m) < A164514(n) for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255437, A164514.

a255527 = (+ 1) . fromJust . (`elemIndex` a255437_list) . a164514

A255507 First differences of A255437.

Original entry on oeis.org

1, 1, -2, 2, 2, 1, 1, 1, -7, 2, 2, 5, 1, 1, 1, 1, 1, -14, 2, 2, 2, 10, 1, 1, 1, 1, 1, 1, 1, -23, 2, 2, 2, 2, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, -34, 2, 2, 2, 2, 2, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -47, 2, 2, 2, 2, 2, 2, 37, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Mar 23 2015

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255437, A255878.

a255507 n = a255507_list !! (n-1)
a255507_list = zipWith (-) (tail a255437_list) a255437_list

A255508 Partial sums of A255437.

Original entry on oeis.org

1, 3, 6, 7, 10, 15, 21, 28, 36, 37, 40, 45, 55, 66, 78, 91, 105, 120, 121, 124, 129, 136, 153, 171, 190, 210, 231, 253, 276, 300, 301, 304, 309, 316, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 631, 634, 639, 646, 655, 666, 703, 741, 780, 820, 861

Offset: 1

Reinhard Zumkeller, Mar 23 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255437, A255879.

a255508 n = a255508_list !! (n-1)
a255508_list = scanl1 (+) a255437_list

A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.

Original entry on oeis.org

1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529

Offset: 1

N. J. A. Sloane, R. K. Guy, Dec 12 1974

These are Hogben's central polygonal numbers
2
.P
3 n
Also the sum of three consecutive triangular numbers (A000217); i.e., a(4) = 19 = T4 + T3 + T2 = 10 + 6 + 3. - Robert G. Wilson v, Apr 27 2001
For k>2, Sum_{n=1..k} a(n) gives the sum pertaining to the magic square of order k. E.g., Sum_{n=1..5} a(n) = 1 + 4 + 10 + 19 + 31 = 65. In general, Sum_{n=1..k} a(n) = k*(k^2 + 1)/2. - Amarnath Murthy, Dec 22 2001
Binomial transform of (1,3,3,0,0,0,...). - Paul Barry, Jul 01 2003
a(n) is the difference of two tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-2)(n-1)(n)/6. - Alexander Adamchuk, May 20 2006
Partial sums are A006003(n) = n(n^2+1)/2. Finite differences are a(n+1) - a(n) = A008585(n) = 3n. - Alexander Adamchuk, Jun 03 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Equals (1, 2, 3, ...) convolved with (1, 2, 3, 3, 3, ...). a(4) = 19 = (1, 2, 3, 4) dot (3, 3, 2, 1) = (3 + 6 + 6 + 4). - Gary W. Adamson, May 01 2009
Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
a(n) is the number of triples (w,x,y) having all terms in {0,...,n} and min(w+x,x+y,y+w) = max(w,x,y). - Clark Kimberling, Jun 14 2012
a(n) = number of atoms at graph distance <= n from an atom in the graphite or graphene network (cf. A008486). - N. J. A. Sloane, Jan 06 2013
In 1826, Shiraishi gave a solution to the Diophantine equation a^3 + b^3 + c^3 = d^3 with b = a(n) for n > 1; see A226903. - Jonathan Sondow, Jun 22 2013
For n > 1, a(n) is the remainder of n^2 * (n-1)^2 mod (n^2 + (n-1)^2). - J. M. Bergot, Jun 27 2013
The equation A000578(x) - A000578(x-1) = A000217(y) - A000217(y-2) is satisfied by y=a(x). - Bruno Berselli, Feb 19 2014
A242357(a(n)) = n. - Reinhard Zumkeller, May 11 2014
A255437(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2015
The first differences give A008486. a(n) seems to give the total number of triangles in the n-th generation of the six patterns of triangle expansion shown in the link. - Kival Ngaokrajang, Sep 12 2015
Number of binary shuffle squares of length 2n which contains exactly two 1's. - Bartlomiej Pawlik, Sep 07 2023
The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered 12-gonal numbers, or centered dodecagonal numbers A003154(n). - Peter M. Chema, Dec 20 2023

			From _Seiichi Manyama_, Aug 12 2017: (Start)
a(1) = 1:
      *
     / \
    /   \
   /     \
  *-------*
.................................................
a(2) = 4:
            *
           / \
          /   \
         /     \
        *---*---*
           / \
      *   /   \   *
     / \ /     \ / \
    /   *-------*   \
   /     \     /     \
  *-------*   *-------*
.................................................
a(3) = 10:
                  *
                 / \
                /   \
               /     \
              *---*---*
                 / \
            *   /   \   *
           / \ /     \ / \
          /   *---*---*   \
         /     \ / \ /     \
        *---*---*   *---*---*
           / \ /     \ / \
      *   /   *---*---*   \   *
     / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   \
   /     \     /     \     /     \
  *-------*   *-------*   *-------*
.................................................
a(4) = 19:
                        *
                       / \
                      /   \
                     /     \
                    *---*---*
                       / \
                  *   /   \   *
                 / \ /     \ / \
                /   *---*---*   \
               /     \ / \ /     \
              *---*---*   *---*---*
                 / \ /     \ / \
            *   /   \---*---*   \   *
           / \ /     \ / \ /     \ / \
          /   *---*---*   *---*---*   \
         /     \ / \ /     \ / \ /     \
        *---*---*   *---*---*   *---*---*
           / \ /     \ / \ /     \ / \
      *   /   *---*---*   *---*---*   \   *
     / \ /     \ / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   *-------*   \
   /     \     /     \     /     \     /     \
  *-------*   *-------*   *-------*   *-------*
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv:1510.08036 [math.CO], 2015.
Jarosław Grytczuk, Bartłomiej Pawlik, and Mariusz Pleszczyński, Variations on shuffle squares, arXiv:2308.13882 [math.CO], 2023. See p. 11.
F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
Milan Janjic, Two Enumerative Functions
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Kival Ngaokrajang, Illustration of triangles expansion
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Leo Tavares, Illustration: Triple Triangles
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Centered Triangular Number
Julie Zhang, Noah A. Rosenberg, and Julia A. Palacios, The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains, arXiv:2506.10856 [math.PR], 2025. See p. 33.
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A001263, A001844, A002061, A003154, A006003 (partial sums), A008486, A008585 = first differences, A045943, A134482, A146325, A226903, A242357, A255437.

a005448 n = 3 * n * (n - 1) `div` 2 + 1
a005448_list = 1 : zipWith (+) a005448_list [3, 6 ..]
-- Reinhard Zumkeller, Jun 20 2013

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..60]]; // Vincenzo Librandi, Sep 13 2015

A005448 := n->(3*(n-1)^2+3*(n-1)+2)/2: seq(A005448(n), n=1..100);
A005448 := -(1+z+z**2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation for offset 0

FoldList[#1 + #2 &, 1, 3 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Join[{1,4},Total/@Partition[Accumulate[Range[50]],3,1]] (* Harvey P. Dale, Aug 17 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 4, 10}, 50] (* Vincenzo Librandi, Sep 13 2015 *)
Table[ j! Coefficient[Series[Exp[x]*(1 + 3 x^2/2)-1, {x, 0, 20}], x, j], {j, 0, 20}] (* Nikolaos Pantelidis, Feb 07 2023 *)
3#+1&/@Accumulate[Range[0,50]] (* Harvey P. Dale, Nov 20 2024 *)

{a(n)=3*(n^2-n)/2+1} /* Michael Somos, Sep 23 2006 */

isok(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k)); \\ Michel Marcus, May 20 2020

Expansion of x*(1-x^3)/(1-x)^4.
a(n) = C(n+3, 3)-C(n, 3) = C(n, 0)+3*C(n, 1)+3*C(n, 2). - Paul Barry, Jul 01 2003
a(n) = 1 + Sum_{j=0..n-1} (3*j). - Xavier Acloque, Oct 25 2003
a(n) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - Lekraj Beedassy, Nov 05 2005
Euler transform of length 3 sequence [4, 0, -1]. - Michael Somos, Sep 23 2006
a(1-n) = a(n). - Michael Somos, Sep 23 2006
a(n) = binomial(n+1,n-1) + binomial(n,n-2) + binomial(n-1,n-3). - Zerinvary Lajos, Sep 03 2006
Row sums of triangle A134482. - Gary W. Adamson, Oct 27 2007
Narayana transform (A001263) * [1, 3, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=4, a(3)=10. - Jaume Oliver Lafont, Dec 02 2008
a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. - Omar E. Pol, Dec 27 2008
a(n) = a(n-1) + 3*n-3. - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} 1/a(n) = A306324. - Ant King, Jun 12 2012
a(n) = 2*a(n-1) - a(n-2) + 3. - Ant King, Jun 12 2012
a(n) = A101321(3,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: -1 + (2 + 3*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016
a(n) = A002061(n) + A000217(n-1). - Bruce J. Nicholson, Apr 20 2017
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 5*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 5/(2*e) - 1. (End)
a(n) = A000326(n) - n + 1. - Charlie Marion, Nov 21 2020

A068722 Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n.

Original entry on oeis.org

1, 35, 247, 925, 2501, 5551, 10795, 19097, 31465, 49051, 73151, 105205, 146797, 199655, 265651, 346801, 445265, 563347, 703495, 868301, 1060501, 1282975, 1538747, 1830985, 2163001, 2538251, 2960335, 3432997, 3960125, 4545751, 5194051, 5909345, 6696097, 7558915

Offset: 0

R. H. Hardin, Feb 26 2002

Conjecture: A255437(a(n)) = 2*n+1, i.e. a(n) = gives the position of the first occurrence of 2*n+1 in A255437. - Reinhard Zumkeller, Mar 23 2015

			Sample flows (. represents a space):
Numbers in long rows are on cell walls showing velocity rightward.
Numbers in long columns are on cell floors showing velocity downwards.
3 X 3 cell centers are at the intersection of long rows and long columns.
n=1:
.. 0 . 0 . 0
.0. -1. -1 . 0
.. 1 . 0. -1
.0 . 0 . 0 . 0
.. 1 . 0. -1
.0 . 1 . 1 . 0
.. 0 . 0 . 0
n=2:
.. 0 . 0 . 0
.0. -2. -1 . 0
.. 2. -1. -1
.0 . 0. -1 . 0
.. 2 . 0. -2
.0 . 2 . 2 . 0
.. 0 . 0 . 0

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. 2 X 2=1, 3, 5, 7..., 4 X 4 A068723, 5 X 5 A068724, 6x6 A068725, by velocity limit 1..13 A068726-A068738.

Cf. A016754, A255437.

a068722 n = (1 + 2 * n + 2 * n ^ 2) * (1 + 3 * n + 3 * n ^ 2)
-- Reinhard Zumkeller, Mar 23 2015

a(n)=(1+2*n+2*n^2)*(1+3*n+3*n^2) \\ Charles R Greathouse IV, Apr 08 2016

a(n) = (1+2*n+2*n^2) * (1+3*n+3*n^2).
G.f.: (1+30*x+82*x^2+30*x^3+x^4)/(1-x)^5. - Colin Barker, Jul 30 2012
E.g.f.: exp(x)*(1 + 34*x + 89*x^2 + 48*x^3 + 6*x^4). - Stefano Spezia, Mar 10 2024

Formula corrected by Colin Barker, Jul 30 2012

A256188 In positive integers: replace k*(k+1)/2 with the first k numbers.

Original entry on oeis.org

1, 2, 1, 2, 4, 5, 1, 2, 3, 7, 8, 9, 1, 2, 3, 4, 11, 12, 13, 14, 1, 2, 3, 4, 5, 16, 17, 18, 19, 20, 1, 2, 3, 4, 5, 6, 22, 23, 24, 25, 26, 27, 1, 2, 3, 4, 5, 6, 7, 29, 30, 31, 32, 33, 34, 35, 1, 2, 3, 4, 5, 6, 7, 8, 37, 38, 39, 40, 41, 42, 43, 44, 1, 2, 3, 4

Offset: 1

Reinhard Zumkeller, Mar 26 2015

nonn

a(A002061(n)) = 1;
a(A253169(n)) = n and a(m) != n for m < A253169(n);
a(A000537(n)) = A000217(n) and a(m) != A000217(n) for m < A000537(n);
see A004202 and A014132 for record values greater than 1 and where they occur: A014132(n) = a(A004202(n)).

			.  A000217 | 1,  3,      6,          10,                 15,       . . .
.  A000027 | _,2,___,4,5,_____,7,8,9,_______,11,12,13,14,_________,16,...
.  A002260 | 1,  1,2,    1,2,3,      1,2,3,4,            1,2,3,4,5,
.  --------+-------------------------------------------------------------
.     a(n) | 1,2,1,2,4,5,1,2,3,7,8,9,1,2,3,4,11,12,13,14,1,2,3,4,5,16,17,...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255437, A000217, A014132, A002260, A000537, A004202, A014132, A002061, A255878 (first differences), A255879 (partial sums).

a256188 n = a256188_list !! (n-1)
a256188_list = f 0 [1..] a002260_tabl where
   f k xs (zs:zss) = us ++ zs ++ f (k + 1) vs zss
                     where (us, v:vs) = splitAt k xs

Table[If[OddQ[Sqrt[8n+1]],Range[(Sqrt[8n+1]-1)/2],n],{n,50}]//Flatten (* Harvey P. Dale, Jun 01 2019 *)

A164514 1 followed by numbers that are not squares.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Jaroslav Krizek, Aug 14 2009

Complement of A000290 for n >= 1.

a(n) = A255437(A255527(n)) and A255437(m) < a(n) for m < A255527(n), i.e. record values in A255437.

Cf. A000037, A255437, A255527.

a164514 n = a164514_list !! (n-1)
a164514_list = 1 : a000037_list
-- Reinhard Zumkeller, Mar 23 2015

is(n)=!issquare(n) || n==1 \\ Charles R Greathouse IV, Sep 01 2015

{1} Union A000037.

a(n) = A000037(n-1) = n-1+floor(1/2 + sqrt(n - 1)) = n-1 + floor( sqrt(n-1 + floor( sqrt(n - 1) ))) for n > 1.

Edited by R. J. Mathar, Aug 21 2009

A255878 First differences of A256188.

Original entry on oeis.org

1, -1, 1, 2, 1, -4, 1, 1, 4, 1, 1, -8, 1, 1, 1, 7, 1, 1, 1, -13, 1, 1, 1, 1, 11, 1, 1, 1, 1, -19, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, -26, 1, 1, 1, 1, 1, 1, 22, 1, 1, 1, 1, 1, 1, -34, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, -43, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Mar 26 2015

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256188, A255437.

a255878 n = a255878_list !! (n-1)
a255878_list = zipWith (-) (tail a256188_list) a256188_list

A255879 Partial sums of A256188.

Original entry on oeis.org

1, 3, 4, 6, 10, 15, 16, 18, 21, 28, 36, 45, 46, 48, 51, 55, 66, 78, 91, 105, 106, 108, 111, 115, 120, 136, 153, 171, 190, 210, 211, 213, 216, 220, 225, 231, 253, 276, 300, 325, 351, 378, 379, 381, 384, 388, 393, 399, 406, 435, 465, 496, 528, 561, 595, 630

Offset: 1

Reinhard Zumkeller, Mar 26 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256188, A255437.

a255879 n = a255879_list !! (n-1)
a255879_list = scanl1 (+) a256188_list

Accumulate[Table[If[OddQ[Sqrt[8n+1]],Range[(Sqrt[8n+1]-1)/2],n],{n,50}]// Flatten] (* Harvey P. Dale, Jun 01 2020 *)

cp's OEIS Frontend

A255527 Where records occur in A255437.

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A255507 First differences of A255437.

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A255508 Partial sums of A255437.

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A005448 Centered triangular numbers: a(n) = 3*n*(n-1)/2 + 1.

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A068722 Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n.

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A256188 In positive integers: replace k*(k+1)/2 with the first k numbers.

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Mathematica

A164514 1 followed by numbers that are not squares.

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A255878 First differences of A256188.

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A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.