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 A253145 Triangular numbers (A000217) omitting the term 1. 2
 0, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The full triangle of the inverse Akiyama-Tanigawa transform applied to (-1)^n*A062510(n)=3*(-1)^n*A001045(n) yielding a(n) is 0,      3,    6,  10,   15,  21,  28, 36, ... -3,    -6,  -12, -20,  -30, -42, -56, ...    essentially -A002378 3,     12,   24,  40,   60,  84, ...         essentially  A046092 -9,   -24,  -48, -80, -120, ...              essentially -A033996 15,    48,   96, 160, ... -33,  -96, -192, ... 63,   192, ... -129, ... etc. First column: (-1)^n*A062510(n). The following columns are multiples of A122803(n)=(-2)^n. See A007283(n), A091629(n), A020714(n+1), A110286, A175805(n), 4*A005010(n). An autosequence of the first kind is a sequence whose main diagonal is A000004 = 0's. b(n) = 0, 0 followed by a(n) is an autosequence of the first kind. The successive differences of b(n) are 0,   0,  0, 3, 6, 10, 15, 21, ... 0,   0,  3, 3, 4,  5,  6,  7, ...  see A194880(n) 0,   3,  0, 1, 1,  1,  1,  1, ... 3,  -3,  1, 0, 0,  0,  0,  0, ... -6,  4, -1, 0, 0,  0,  0,  0, ... 10, -5,  1, 0, 0,  0,  0,  0, ... -15, 6, -1, 0, 0,  0,  0,  0, ... 21, -7,  1, 0, 0,  0,  0,  0, ... The inverse binomial transform (first column) is the signed sequence. This is general. Also generalized hexagonal numbers without 1. - Omar E. Pol, Mar 23 2015 LINKS Muniru A Asiru, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA Inverse Akiyama-Tanigawa transform of (-1)^n*A062510(n). a(n) = (n+1)*(n+2)/2 for n > 0. - Charles R Greathouse IV, Mar 23 2015 a(n+1) = 3*A001840(n+1) + A022003(n). a(n) = A161680(n+2) for n >= 1. - Georg Fischer, Oct 30 2018 MATHEMATICA Prepend[Table[(n + 1) (n + 2)/2, {n, 49}], 0] (* Michael De Vlieger, Mar 23 2015 *) PROG (PARI) a(n)=if(n, (n+1)*(n+2)/2, 0) \\ Charles R Greathouse IV, Mar 23 2015 (GAP) Concatenation([0], List([1..50], n->(n+1)*(n+2)/2)); # Muniru A Asiru, Oct 31 2018 CROSSREFS Cf. A000217, A179865, A001045, A062510, A002378, A046092, A033996, A122803, A007283, A091629, A020714, A110286, A161680, A175805, A005010, A194880, A001840, A022003, A255935. Sequence in context: A179865 A105339 A089594 * A161680 A000217 A105340 Adjacent sequences:  A253142 A253143 A253144 * A253146 A253147 A253148 KEYWORD nonn,easy AUTHOR Paul Curtz, Mar 23 2015 STATUS approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)